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As Wikipedia explains, one photon passing through a crystal sometimes down-converts to two photons. Wikipedia says total energy and momentum are conserved by just considering the three photon states; is Wikipedia wrong here?

It seems a phonon (or something else) is needed too. If Wikipedia is right, can you provide 3 example (non-parallel) momenta vectors so that I can see my logic mistake?

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The crystal itself breaks translation symmetry and absorbs a tiny amount of momentum as a whole during a seeming momentum violating process. This is because the phenomenon is coherent along all the crystal atoms, it's not paradoxical. –  Ron Maimon Sep 7 '12 at 15:28
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I'm not sure why you don't want the momenta vectors to be parallel. Normally they are parallel in parametric downconversion.

That doesn't solve the problem, because the parametric downconversion happens in a material, and materials always have dispersion (different refractive index at different wavelengths). The nature of dispersion makes it difficult in normal circumstances to simultaneously have $\omega = \omega_1 + \omega_2$ and $k = k_1 + k_2$, even when the wavevectors are parallel. But with a bit of cleverness and effort it is possible.

This field of knowledge is called PHASE MATCHING. It is a basic and important topic in nonlinear optics. In a nonlinear optics textbook, it would normally be discussed in the first chapter. I doubt I would do it justice in a few sentences.

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Wikipedia says they're usually off-parallel for this interesting case: en.wikipedia.org/wiki/Spontaneous_parametric_down_conversion Whether we talk about conservation of crystal momentum (for a bulk infinite crystal) or mechanical momentum (for finite crystal comparing global state before and after crystal passage), I don't get how that first Wikipedia figure can be true. –  bobuhito Sep 7 '12 at 2:42
    
Or, maybe crystal momentum is what is being drawn and dispersion is the reason (the two emitted photons are moving slower than the original photon, so their energies are less than I had thought)? –  bobuhito Sep 7 '12 at 2:45
    
You, the experimenter, are free to decide whether they are parallel or not. You decide based on whatever makes the experiment work best. If you want it to be parallel, you make it phase-matched for parallel waves. If you want it to be non-parallel, you make it phase-matched for non-parallel waves. –  Steve B Sep 7 '12 at 19:33
    
Phase matching is NOT really related to conservation of momentum. With phase-matching, the waves add in phase to become very strong. Without phase-matching, the waves add with random phase offsets to give a small (but nonzero) total. The fact that nonlinearly-generated light goes primarily in phase-matched directions is analogous to the fact that light bouncing off a diffraction grating goes primarily in specific directions. –  Steve B Sep 7 '12 at 19:39
    
Sorry, I think you have taken my question differently than intended. The 3 momenta are not inputs. I only get to input one momentum. The other 2 momenta are outputs. –  bobuhito Sep 7 '12 at 21:36
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Parametric Down-Conversion (PDC) is a parametric process[1]. By definition of a parametric process the initial and final state of crystal/medium remains same, hence the energy is, by default, conserved (once you prove that the Down-conversion is actually parametric in nature). The momentum conservation is a condition that impose on this down-conversion since it helps improve the efficiency of conversion. So yes the crystal-momentum is only comes into play if the momentum conservation is can not be satisfied perfectly for the PDC to take place.

I do not understand what you mean by 3-examples, but I will give a try. In crystals with normal dispersion relation, i.e. refractive index decreases as wavelength increases, one can not satisfy the phase-matching (PM) condition[2]. In order to satisfy the PM conditions one employs the birefringence. (It is a long and very easy analysis) so one can show that in case of birefringent crystal with normal dispersion can only show 2-types of phase-matching conditions (energy conservation is a invariant condition in 2-types).

$$ (\bf{k_p})_{fast} = (\bf{k_s})_{slow} +(\bf{k_i})_{slow} \ \ Type-I \\ (\bf{k_p})_{fast} = (\bf{k_s})_{slow} +(\bf{k_i})_{fast} \ \ Type-II \\ (\bf{k_p})_{fast} = (\bf{k_s})_{fast} +(\bf{k_i})_{slow} \ \ Type-II \\ $$

Fast slow only implies you need to use smaller refractive index of the 2-refractive indices found in birefringent crystals (e.g. for negative uniaxial crystals fast implies extraordinary and slow means ordinary).

One last comment on your question. Although collinear phase-matching is possible and can be achieved in lab, one only want to satisfy the noncollinear phase-matching to condition for following two reasons (there might be more but these readily come to mind).

  1. In collinear PM, pump also comes out along with signal and idler owing to the fact efficiency of PDC is <100%.
  2. One usually want two entangled systems separated in space, so you could do experiment related to entanglement and eventually create some related e.g. Quantum Teleportation device.

[1] http://en.wikipedia.org/wiki/Parametric_process_(optics)

[2] Robert W. Boyd, Nonlinear Optics, 3rd Edition, section 2.3


The following is extra-bits I have to say.

You have to understand that PDC is only observed in nonlinear crystals. You will not get observe this effect in usual linear materials e.g. glass (though it is a theoretical possibility if the intensity is absurdly high).

The wiki article has title Spontaneous-PDC, this is because there also something known Stimulated-PDC (I do not know the reason why it is not used for PDC, may be because it is little difficult since there are 4-waves involved two-of-which have the same frequency).

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No matter what kp, ks, and ki you choose (by "example", I wanted some actual numbers), I don't think that you can conserve both energy and momentum. Instead, I think Ron Maimon is correct in his comment above. So, I'd say Wikipedia is wrong (in answer to my original question) and "parametric" is not used/defined well here. –  bobuhito Sep 7 '13 at 16:39
    
dude look up the wiki article for parametric process. I just meant that parametric ==> you don't need to take into account for the crystal momentum. Though I do not know why crystal momentum is not included in PDC phase-matching, I do know that you can satisfy PM conditions and calculate the non-collinear and collinear ks and ki. In this image you will find signal and idler ring on the screen at the distance = 10 cm from BBO crystal with pump wavelength = 351.1 nm for $\theta_{pump} = 47.4^\circ$. See this –  The Imp Sep 8 '13 at 0:06
    
I guess KDN answers related question very well. –  The Imp Sep 8 '13 at 1:01
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